Representations of Knot Groups and Twisted Alexander Polynomials

By Xiaoxia Lin. 2001

240 citations recorded by the source corpus.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/38403dc65469c8e92df2b49cc94bb234bce19476

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Non Abelian Twisted Reidemeister Torsion for Fibered Knots (2004, background): “In the Main Theorem, we restrict our attention on the irreducible non-metabelian representations of GK in G. This restriction is just technical; moreover, there exists only a finite number of irreducible metabelian representations of GK in SU(2) (see [ Lin01 , Proposition 4.2]).”
Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation (2008, methodology): “The early development of this invariant as a tool in classical knot theory, in which case X was taken to be a classical knot complement, appeared in such papers as [18,24,27,41].”
The Thurston norm and twisted Alexander polynomials (2012, methodology): “This invariant was first introduced by Lin [Lin01] and Wada [Wa94] using slightly different normalizations.”
A Survey of Classical Knot Concordance (2003, methodology): “ of this module. Although this three-dimensional approach does not give the signature invariant, it has the advantage of being completely algorithmic in computation, via a procedure ﬁrst developed in [66, 112] and applied in [56, 57, 58]. A computer implementation of that algorithm facilitated the c”
A SURVEY OF TWISTED ALEXANDER POLYNOMIALS (2009, methodology): “The first approach was used in Lin’s original paper [Lin01] to distinguish knots with the same Alexander module.”
Fibred knots and twisted Alexander invariants (2001, unknown): “Since a twisted version of the Alexander polynomial of a knot was first defined by Lin [ 14 ], the twisted Alexander invariants have been studied by several authors, including Wada [18], Jiang and Wang [9], and Kitano [12].”
Cocycle knot invariants from quandle modules and generalized quandle homology (2003, methodology): “In [31, 41] group representations of knot groups are used to d efine twisted Alexander polynomials.”
Twisted Alexander polynomials detect fibered 3-manifolds (2008, methodology): “These were introduced, for the case of knots, by Xiao-Song Lin in 1990 (published in [ Li01 ]), and his definition was later generalized to 3–manifolds by Wada [Wa94], Kirk–Livingston [KL99] and Cha [Ch03].”
Reidemeister torsion, twisted Alexander polynomial and fibered knots (2003, unknown): “The notion of Alexander polynomials twisted by a representation and its applications have appeared in several papers (see [2], [6], [8], [18]).”
A vanishing theorem for twisted Alexander polynomials with applications to symplectic 4-manifolds (2012, methodology): “ty or toroidal boundary. 2. Definition of twisted Alexanderpolynomials In this section we quickly recall the deﬁnition of twisted Alexander polynomials. This invariant was initially introduced by Lin [Li01], Wada [Wa94] and Kirk– Livingston [KL99]. We refer to [FV10a] for a detailed presentation. Le”
The Thurston norm, fibered manifolds and twisted Alexander polynomials (2005, methodology): “The twisted Alexander polynomial of a knot was introduced by Lin [19] in 1990.”